Finite Math Examples

$r = \sqrt{r^{2} - 3 r - 9}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{r^{2} - 3 r - 9} = r$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{r^{2} - 3 r - 9}}^{2} = r^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{r^{2} - 3 r - 9}$ as ${(r^{2} - 3 r - 9)}^{\frac{1}{2}}$ .

${({(r^{2} - 3 r - 9)}^{\frac{1}{2}})}^{2} = r^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(r^{2} - 3 r - 9)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(r^{2} - 3 r - 9)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(r^{2} - 3 r - 9)}^{\frac{1}{2} \cdot 2} = r^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(r^{2} - 3 r - 9)}^{\frac{1}{2} \cdot 2} = r^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(r^{2} - 3 r - 9)}^{1} = r^{2}$

Step 3.2.1.2

Simplify.

$r^{2} - 3 r - 9 = r^{2}$

Step 4

Solve for $r$ .

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Step 4.1

Move all terms containing $r$ to the left side of the equation.

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Step 4.1.1

Subtract $r^{2}$ from both sides of the equation.

$r^{2} - 3 r - 9 - r^{2} = 0$

Step 4.1.2

Combine the opposite terms in $r^{2} - 3 r - 9 - r^{2}$ .

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Step 4.1.2.1

Subtract $r^{2}$ from $r^{2}$ .

$- 3 r - 9 + 0 = 0$

Step 4.1.2.2

Add $- 3 r - 9$ and $0$ .

$- 3 r - 9 = 0$

Step 4.2

Add $9$ to both sides of the equation.

$- 3 r = 9$

Step 4.3

Divide each term in $- 3 r = 9$ by $- 3$ and simplify.

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Step 4.3.1

Divide each term in $- 3 r = 9$ by $- 3$ .

$\frac{- 3 r}{- 3} = \frac{9}{- 3}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $- 3$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{- 3 r}{- 3} = \frac{9}{- 3}$

Step 4.3.2.1.2

Divide $r$ by $1$ .

$r = \frac{9}{- 3}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $9$ by $- 3$ .

$r = - 3$

Step 5

Exclude the solutions that do not make $r = \sqrt{r^{2} - 3 r - 9}$ true.

$No solution$